Not in our tests, and nobody in our beta test period had any comments. The logic around controlling the connection is pretty intelligent in what it does.

A connection that's open isn't problematic at all, assuming it's being used - lots of setups/teardowns can be an issue (which is why polling apps will consistently take up a large portion of your battery life). The main issue is if the app is using a radio when it shouldn't be. Luckily, we can control this pretty well with our handling code and have been able to balance out needs pretty well between delivering a message as quickly as possible and not destroying battery life.

If you're an Android dev, we'd love your feedback on it in general :)